We have a dynamical system: \begin{align*} \dot{x}&=Ax+Bu\\ x(0)&=x_{0} \end{align*} that is controllable. Given the cost function: $$ J(u,t_{1})=\int_{0}^{t_{1}}u^{T}udt,\quad x(t_1)=0, $$ it is known that the optimal control is $u^{*}=-B^{T}e^{A^{T}(t_{1}-t)}W(0,t_{1})^{-1}e^{At_{1}}x_{0}$. How can I show that we can write $u^{*}$ on the form: $$ u^{*}=-B^{T}e^{A^{T}(t_{1}-t)}W(t,t_{1})^{-1}e^{A(t_{1}-t)}x(t). $$ My attempted solution so far is the following.
Given the optimal control we rewrite: \begin{align*} u^{*}&=-B^{T}e^{A^{T}(t_{1}-t)}W(0,t_{1})^{-1}e^{At_{1}}x_{0}\\ &=-B^{T}e^{A^{T}(t_{1}-t)}W(0,t_{1})^{-1}e^{A(t_{1}-t)}e^{At}x_{0}\\ &=-B^{T}e^{A^{T}(t_{1}-t)}W(0,t_{1})^{-1}e^{A(t_{1}-t)}\left(x(t)-\int_{0}^{t}e^{A(t-s)}Bu(s)ds\right)\\ &=-B^{T}e^{A^{T}(t_{1}-t)}W(0,t_{1})^{-1}e^{A(t_{1}-t)}x(t)+B^{T}e^{A^{T}(t_{1}-t)}W(0,t_{1})^{-1}e^{A(t_{1}-t)}\int_{0}^{t}e^{A(t-s)}Bu(s)ds \end{align*} where I used that the solution to a system on the given form is: $$ x(t)=e^{At}x_{0}+\int_{0}^{t}e^{A(t-s)}Bu(s)ds. $$ The first term in my attempted solution looks close to the desired form but I am not sure how to continue from here, or if I am completely off.
Furthermore, a similar question to mine is posted here: Linear Quadratic optimal control in feedback form but I do not know how to tie it together to my desired form.